= Solution
\b[No <weakly stationary process> solves the model with nondegenerate noise.] Iteration would give
$$
X_t-X_{t-m}=\sum_{j=0}^{m-1}\varepsilon_{t-j},\qquad \operatorname{Var}(X_t-X_{t-m})=m\sigma^2.
$$
For a <weakly stationary process> with finite <variance> $V$, the <Cauchy-Schwarz inequality> instead gives $\operatorname{Var}(X_t-X_{t-m})=2V-2\gamma(m)\leq4V$. These two bounds contradict each other for large $m$. This argument does not assume that $X_{t-m}$ is independent of the intervening noise, so it excludes noncausal solutions too. The corresponding <unit-root autoregressive process> has persistent <random walk> behavior rather than stationary fluctuations.
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